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An air-traffic controller observes two aircraft on his radar screen. The first is at altitude 800...

Question

An air-traffic controller observes two aircraft on his radar screen. The first is at altitude 800...

An air-traffic controller observes two aircraft on his radar screen. The first is at altitude 800 m, horizontal distance 19.2 km, and 25.0° south of west. The second aircraft is at altitude 1 100 m, horizontal distance 17.6 km, and 20.0° south of west. What is the distance between the two aircraft? (Place the x axis west, the y axis south, and the z axis vertical.)

Answers

300 mm 2265 m 800 m 800 m 2265 m Horizontal Plane parallel to Ground Vertical Plane

Left side figure shows the position of airplanes A and B detected by air-traffic controller observing from O.

We write position vector of A as, \underset{OA}{\rightarrow} = 19.2 cos25 \hat{i} + 19.2 sin25 \hat{j} km

We write position vector of B as, \underset{OB}{\rightarrow} = 17.6 cos20 \hat{i} + 17.6 sin20 \hat{j} km

Vector \underset{AB}{\rightarrow} = \underset{OB}{\rightarrow} - \underset{OA}{\rightarrow} = (17.6 cos20 - 19.2 cos25) \hat{i} + (17.6 sin20 - 19.2 sin25 ) \hat{j} km

\underset{AB}{\rightarrow} = -0.863 \hat{i} -2.095 \hat{j} km

Distance between A and B = | \underset{AB}{\rightarrow} | = ( 0.863 \times 0.863 + 2.095 \times 2.095 )1/2 = 2.265 km

Right side figure shows the vertical plane , where the air planes height from ground are seen.

Distance between airplanes A and B in space = ( 2265 \times 2265 + 300 \times 300 )1/2 = 2285 m


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